Electrostatic cumulation of high-current electron beams for terahertz sources

The electrostatic cumulation of current density in relativistic vacuum diodes with ring-type cathodes is described theoretically and confirmed experimentally. The distinctive feature of the suggested cumulation mechanism is a very low energy spread of electrons. As a result of electrostatic cumulation, a thin relativistic electron beam with a current density of 1 kA/mm$^2$ can be formed. This quantity exceeds typical current density values in high-current Cherenkov sources by an order of magnitude. Such a beam can be used as an active medium in high-power terahertz sources.


I. INTRODUCTION
Generation of high-power radiation is one of the main goals of vacuum terahertz electronics [1,2]. The progress in this field is strongly connected with production in highcurrent accelerators [1][2][3] of relativistic electron beams with ever-increasing density .
A typical high-current accelerator for a THz source consists of an axially symmetric cathode and anode [3]. The cathode is hemispherical with curvature radius of several centimeters and it is spaced tens of millimeters from the concave anode. The accelerator produces about 1 kA of total current at ∼ 1 MV applied voltage. The hole in the anode cuts a small central part of the accelerated beam (∼ 10 A), which is guided by magnetic field with the strength ∼ 1 T. Low efficiency of beam usage limits applications of such an approach.
The second disadvantage of applying traditional highcurrent accelerators is high value of an external guiding field (1 T) that results in large dimensions of the whole system. High magnetic fields are necessary to prevent beam expansion due to the Coulomb repulsion. Meanwhile an alternative mechanism precluding beam expansion is already well-known. This mechanism doesn't require external guiding fields. It is based on magnetic cumulation discovered by Bennett [4]. Magnetic cumulation provides charged-particle beam intensities as high as 10 GW/mm 2 [5,6], thus enabling the laboratory investigation of the extreme state of matter. However, the large particle energy spread does not enable to use magnetically cumulated electron beams in terahertz radiation sources.
This paper considers one more alternative for highcurrent electron beam cumulation, namely electrostatic cumulation [7,8]. Occurring in a relativistic vacuum diode with a ring-type cathode, this mechanism doesn't suffer from particle energy spread [8]. Our principle task is to provide theoretical description of cumulation mechanism and the experimental verification thereof. We will show that electrostatic cumulation provides current densities (∼ 1 kA/mm 2 ) sufficient to seed high-power terahertz radiation sources.

II. THE PHENOMENON OF ELECTROSTATIC CUMULATION
Electrostatic cumulation was first revealed during modeling of high-current accelerators [8]. The qualitative picture of electrostatic cumulation can be described as follows. In a relativistic vacuum diode, electron emission is most intense from the cathode's edges (Fig. 1). Let us consider electrons emitted from the inner edge. The Coulomb repulsion causes the charged particles to rush to the region free from the beam. As a result, the accelerated motion of electrons toward the anode comes alongside the radial motion to the cathode's symmetry axis. As a result, the high-current beam density increases multifold on the axis as compared to the average current density in the cathode-anode gap [7,8].  spot on the anode at the moment corresponding to the maximum accelerating voltage is as large as 10 A/mm2, being 5 times greater than the average current density of the high-current diode. A typical radius of the spot is about 1 mm. Thus, the simulation result indicates the electron-beam cumulation on the axis of a high-current diode with a ring-type cathode.
The undeniable advantage of electrostatic cumulation is a very low energy spread of particles due to the laminar flow of charged particles. In contrast, self-focusing of a beam by its own magnetic field leads to a turbulent flow, and the charged particles acquire a significant energy spread. In this case, the electron flow is like a com-pressed relativistic gas with the electron temperature T e determined by the accelerating voltage, T e ∼ qeU k B . (Symbol k B denotes the Boltzman constant.) To obtain information about electron beam parameters, we use a nanosecond pulse-periodic electron accelerator with a compact SF6-insulated high-voltage generator providing pulsed voltage up to 400 kV in 30 Ohm resistive load with a full width at half maximum of 130 ns and rise time of 30 ns [7]. To obtain full-sized imprints of electron beams, we use a radiochromic dosimetry film placed 3 mm behind the anode mesh made of stainless steel (the geometrical transparency of the mesh is 0.77); the cathode-anode gap is 20 mm. The dosimetry film enables to obtain information about the total absorbed dose over the beam cross section, caused by passage of charged particles.
Our first experiments had shown that the intense flow of charged particles on the axis had burned the film through. For this reason, in further experiments we placed 70 µm-thick aluminium foil in front of the dosimetry film to decrease the absorbed dose (see Fig. 3) [7]. This enables us to cut off the flows of both the cathode plasma and the weakly-relativistic electrons produced at the voltage pulse decay. The experiments conducted with one, two, and three foils demonstrated that a sharp increase in the absorbed dose remains in the center. This means that the particle flow consists of high-energy electrons at the beam axis. In the experiments with three foil layers cutting off all electrons whose energy was less than 250 keV, the absorbed dose in the center was almost four times as large as the average dose across the beam cross section, showing a good agreement with the simulation results.

III. TRANSPORT OF A DENSE ELECTRON BEAM AT 4 MEV
Let us note here that both the simulations and the experiments were performed at maximum accelerating voltage 400 kV. The estimates show that it is possible to achieve the beam current density more than 1 kA/mm 2 with accelerating voltages of several megavolts. This quantity is several times higher than those obtained in existing high-current terahertz sources producing 100 kW of radiation power. The cumulated electron beam can be extracted with the help of a thin channel in the anode (Fig. 4). The beam may be used in terahertz sources. Cumulated electron beams are explored with the particle-in-cell code described in Appendix A. The accelerator geometry consists of two parts: a cathode and an anode with a channel in its center. The cathode has an outer radius r c of 10 mm. The hole radius r h has a value 8 mm. The cathode-anode gap h ac is 15 mm. The channel radius r ch has values 0.5 or 1 mm. The anode radius r a is 200 mm.
Simulations show that a typical current density in the channel j av is about 1 kA/mm 2 (Fig. 5). The transport length of cumulated beam lies in the millimeter range (Fig. 6). Estimations show that the cumulated electron beam, injected in the slow-wave structure, can lead to the Cherenkov terahertz radiation of 40-400 MW power if we assume 1-10% efficiency, which is a typical value for the high-current Cherenkov generators [3,9].

IV. CONCLUSIONS
In this paper we describe a new cumulation mechanism for high-current beams in relativistic vacuum diodes with a ring-type cathode. The basis of the cumulation mechanism is electrostatic repulsion of electrons emitted from the inner edge of the cathode. At several hundred kilovolts, the electrostatic cumulation is verified experimentally. The current density and beam intensity equal to 1-10 GW/mm 2 and 1 kA/mm 2 , respectively, are obtained in simulations at 4 MV.
A very low particle energy spread in the region of maximum current density is a distinctive feature of the described cumulation mechanism. Due to electrostatic cumulation, a thin relativistic electron flow with a radial width of 1 mm and beam current density of 1 kA/mm 2 is formed. This quantity is several times higher than that obtained in existing high-current terahertz sources producing 0.1-1 MW of radiated power. As a result, electrostatic cumulation can sufficiently increase the radiation output. One of the main advantages of highcurrent terahertz sources with electrostatic cumulation is the absence of external magnetic fields causing large dimensions of the whole system.

Appendix A: Simulation of high-current electron beams
Self-consistent simulation of relativistic motion in high-current diodes is usually performed using the particle-in-cell method [10][11][12][13][14][15][16]. The method is often implemented in a quasi-stationary approximation [10,11]. The approximation is applied when electric and magnetic fields are slowly varying functions of time. As a result, the displacement currents and induction fields can be neglected.
As we are going to deal with axially symmetric geometries the simulation is performed in cylindrical RZcoordinates. So, let us introduce the space mesh as follows: Spatial dimensions of the grid cells are set to be equal, i.e., ∆R = ∆Z. The method implies representation of charged particles' flow as a collection of macroparticles. Each of them contains a large number of real charge carriers. Macroparticles have certain spatial distributions of mass and charge those contribute to the physical quantities (such as current and charge densities) defined on the grid. The latter give us possibility to find electric and magnetic fields in the grid nodes. Interpolation of the grid fields to macroparticles' location determines forces acting on charges. As a result, new particles' locations and velocities can be found. At this point, all the procedures are repeated again. Now, let us proceed to the brief description of the algorithm used in our simulations.
Electric field E is expressed by electrostatic potential φ, The potential φ obeys the Poisson equation, Symbol ρ denotes the charge density.
To solve the Poisson equation, we use the Jacobi iterative method [13], = −4πρ n i,j for i = 0, φ n,0 i,j = φ n−1 i,j , proven to be effective in plasmadynamics and for solving the vacuum electronic problems [8]. Here, s is the iteration number and n is the time step number. Iterations in A6 occur until all the quantities |φ n,s − φ n,s−1 | become less than U . The parameter 1 is a small fraction of accelerating voltage U . (In our simulations, is set to be equal to 10 −5 .) As a rule, iterative methods are slowly converging [13]. However, this is not always the case in dynamical problems [8]. The main reason is proper choice of initial approximation in the iteration scheme. Namely, the potential distribution at the previous time step n − 1 plays the role of initial approximation at the step n. As a result, the Jacobi method reduces to one-three iterations at each time step. The procedure requires much less simulation time than, say, the computation of new velocities and positions of the particles.
To solve the Poisson equation, we use the Dirichlet boundary conditions. The conditions imply specification of potentials at the cathode (φ = Uc) and the anode (φ = Ua) and at the edge of the computational region (z = 0). For the latter, the logarithmic distribution is used [9] φ = Uc + (Ua − Uc) ln(r/rc) ln(ra/rc) .
This distribution exactly describes φ change in the gap between two infinite cylinders. The potential distribution approaches A10 if the edge of the computational region (z = 0) is located at a considerable distance from the electron-emitting surface.
Charge density ρ n i,j can be found with the linear weighting procedure [14]. This means that each particle contributes to the charge density of the four nearest nodes according to the following formulas ρ n i,j = α ∆ρ α,n i,j , Here, ∆V = π∆R 2 ∆Z(2j + 1) for i = 0, qα and r n α = (r n α , z n α ) are particles' charges and positions, respectively.
The electric field is related to electrostatic potential φ n i,j as A non-rotating axially symmetric electron beam possesses only the polar component of the self-induced magnetic field H. The component H can be found using the Stokes theorem: H = 4π r 0 jz(r1, z)r1dr1 cr .
(A8) (To calculate the magnetic field properly, one must include the current in the cathode itself [9,10].) Note that the axial component of the current density can be found by the same weighting procedure as the charge density. It is only necessary to substitute qαv n zα instead of qα in A6. Numerical integration of the Newton-Lorentz equations is the most time-consuming procedure. That is why the integration algorithm must be as fast as possible. During several decades the Boris algorithm was used for this purpose. However, this algorithm can lead to spurious forces acting on particles [17]. To overcome the drawback, a new algorithm for momenta integration has been proposed by J.-L. Vay